Direct Scattering for the Benjamin-Ono Equation with Rational Initial Data

Abstract

We compute the scattering data of the Benjamin-Ono equation for arbitrary rational initial conditions with simple poles. Specifically, we obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient.

Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE

“…Further progress of the IST theory for BO is due to Miller and Wetzel [190] who studied the direct scattering problem of the Fokas-Ablowitz IST theory when the potential is a rational function with simple poles and obtained the explicit formula for the scattering data.…”

“…Using the exact formulas for the scattering data of the BO equation valid for general rational potential with simple poles that they obtained in [190], Miller and Wetzel [189] analyzed rigorously the scattering data in the small dispersion limit, deducing in particular precise asymptotic formulas for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant associated with each eigenvalue on the eigenvalue itself. Such an analysis seems to be unknown for more general potentials.…”

The scattering transform for the Benjamin–Ono equation in the small-dispersion limit

“…Here, the potential u 0 : R → R is the initial condition u(x, 0) = u 0 (x) for (1), and the operator C + : L 2 (R) → H + (R) is the self-adjoint orthogonal projection onto the Hardy space. As first noted in [11] and generalized in [19], the direct scattering problem for the BO equation can be effectively solved for rational potentials of the form…”

“…In this paper we use recently obtained exact formulae [19] to study the asymptotic behavior of the scattering data for the BO equation (1) in the limit → 0. These formulae hold for rational initial data with simple poles, and for convenience we adopt here the further condition of a single local extremum (Definition 3.1).…”

“…

Using exact formulae for the scattering data of the Benjamin-Ono equation valid for general rational potentials recently obtained in [19], we rigorously analyze the scattering data in the small-dispersion limit. In particular, we deduce precise asymptotic formulae for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant (associated with each eigenvalue) on the eigenvalue itself.

…”